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A285321 Square array A(1,k) = A019565(k), A(n,k) = A065642(A(n-1,k)), read by descending antidiagonals.

%I A285321 #24 Mar 27 2021 06:17:18
%S A285321 2,3,4,6,9,8,5,12,27,16,10,25,18,81,32,15,20,125,24,243,64,30,45,40,
%T A285321 625,36,729,128,7,60,75,50,3125,48,2187,256,14,49,90,135,80,15625,54,
%U A285321 6561,512,21,28,343,120,225,100,78125,72,19683,1024
%N A285321 Square array A(1,k) = A019565(k), A(n,k) = A065642(A(n-1,k)), read by descending antidiagonals.
%C A285321 A permutation of the natural numbers > 1.
%C A285321 Otherwise like array A284311, but the columns come in different order.
%H A285321 Antti Karttunen, <a href="/A285321/b285321.txt">Table of n, a(n) for n = 1..120; the first 15 antidiagonals of array</a>
%F A285321 A(1,k) = A019565(k), A(n,k) = A065642(A(n-1,k)).
%F A285321 For all n >= 2: A(A008479(n), A087207(n)) = n.
%e A285321 The top left 12x6 corner of the array:
%e A285321    2,   3,  6,     5,  10,  15,  30,      7,  14,  21,  42,   35
%e A285321    4,   9, 12,    25,  20,  45,  60,     49,  28,  63,  84,  175
%e A285321    8,  27, 18,   125,  40,  75,  90,    343,  56, 147, 126,  245
%e A285321   16,  81, 24,   625,  50, 135, 120,   2401,  98, 189, 168,  875
%e A285321   32, 243, 36,  3125,  80, 225, 150,  16807, 112, 441, 252, 1225
%e A285321   64, 729, 48, 15625, 100, 375, 180, 117649, 196, 567, 294, 1715
%t A285321 a065642[n_] := Module[{k}, If[n == 1, Return[1], k = n + 1; While[ EulerPhi[k]/k != EulerPhi[n]/n, k++]]; k];
%t A285321 A[1, k_] := Times @@ Prime[Flatten[Position[#, 1]]]&[Reverse[ IntegerDigits[k, 2]]];
%t A285321 A[n_ /; n > 1, k_] := A[n, k] = a065642[A[n - 1, k]];
%t A285321 Table[A[n - k + 1, k], {n, 1, 10}, {k, n, 1, -1}] // Flatten (* _Jean-François Alcover_, Nov 17 2019 *)
%o A285321 (Scheme)
%o A285321 (define (A285321 n) (A285321bi (A002260 n) (A004736 n)))
%o A285321 (define (A285321bi row col) (if (= 1 row) (A019565 col) (A065642 (A285321bi (- row 1) col))))
%o A285321 (Python)
%o A285321 from operator import mul
%o A285321 from sympy import prime, primefactors
%o A285321 def a019565(n): return reduce(mul, (prime(i+1) for i, v in enumerate(bin(n)[:1:-1]) if v == '1')) if n > 0 else 1 # This function from _Chai Wah Wu_
%o A285321 def a007947(n): return 1 if n<2 else reduce(mul, primefactors(n))
%o A285321 def a065642(n):
%o A285321     if n==1: return 1
%o A285321     r=a007947(n)
%o A285321     n = n + r
%o A285321     while a007947(n)!=r:
%o A285321         n+=r
%o A285321     return n
%o A285321 def A(n, k): return a019565(k) if n==1 else a065642(A(n - 1, k))
%o A285321 for n in range(1, 11): print([A(k, n - k + 1) for k in range(1, n + 1)]) # _Indranil Ghosh_, Apr 18 2017
%Y A285321 Transpose: A285322.
%Y A285321 Cf. A019565, A065642.
%Y A285321 Cf. A008479 (index of the row where n is located), A087207 (of the column).
%Y A285321 Cf. arrays A284311, A285325, also A285332.
%K A285321 nonn,tabl
%O A285321 1,1
%A A285321 _Antti Karttunen_, Apr 17 2017